A Parameterized Perspective on Protecting Elections
aa r X i v : . [ c s . M A ] M a y A Parameterized Perspective on ProtectingElections
Palash Dey
Indian Institute of Technology, Kharagpur, [email protected]
Neeldhara Misra
Indian Institute of Technology, Gandhinagar, [email protected]
Swaprava Nath
Indian Institute of Technology, Kanpur, [email protected]
Garima Shakya
Indian Institute of Technology, Kanpur, [email protected]
Abstract
We study the parameterized complexity of the optimal defense and optimal attack problems in voting.In both the problems, the input is a set of voter groups (every voter group is a set of votes) and twointegers k a and k d corresponding to respectively the number of voter groups the attacker can attackand the number of voter groups the defender can defend. A voter group gets removed from the electionif it is attacked but not defended. In the optimal defense problem, we want to know if it is possible forthe defender to commit to a strategy of defending at most k d voter groups such that, no matter which k a voter groups the attacker attacks, the outcome of the election does not change. In the optimalattack problem, we want to know if it is possible for the attacker to commit to a strategy of attacking k a voter groups such that, no matter which k d voter groups the defender defends, the outcome ofthe election is always different from the original (without any attack) one. We show that both theoptimal defense problem and the optimal attack problem are computationally intractable for everyscoring rule and the Condorcet voting rule even when we have only 3 candidates. We also show thatthe optimal defense problem for every scoring rule and the Condorcet voting rule is W [ ] -hard for boththe parameters k a and k d , while it admits a fixed parameter tractable algorithm parameterized by thecombined parameter ( k a , k d ) . The optimal attack problem for every scoring rule and the Condorcetvoting rule turns out to be much harder – it is W [ ] -hard even for the combined parameter ( k a , k d ) .We propose two greedy algorithms for the O PTIMAL D EFENSE problem and empirically show that theyperform effectively on reasonable voting profiles.
Keywords and phrases parameterized complexity, election control, optimal attack, optimal defense
The problem of election control asks if it is possible for an external agent, usually with afixed set of resources, to influence the outcome of the election by altering its structure insome limited way. There are several specific manifestations of this problem: for instance,one may ask if it is possible to change the winner by deleting k voter groups, presumablyby destroying ballot boxes or rigging electronically submitted votes. Indeed, several casesof violence at the ballot boxes have been placed on record [7, 2], and in 2010, Halderman A Parameterized Perspective on Protecting Elections and his students exposed serious vulnerabilities in the electronic voting systems that arein widespread use in several states [1]. A substantial amount of the debates around therecently concluded presidential elections in the United States revolved around issues ofpotential fraud, with people voting multiple times, stuffing ballot boxes, etc. all of whichare well recognized forms of election control. For example, Wolchok et al. [54] studiedsecurity aspects on Internet voting systems.
Parameters O
PTIMAL D EFENSE O PTIMAL A TTACK
Scoring rules Condorcet Scoring rules Condorcet k d W [ ] -hard [Theorem 15] W [ ] -hard [Theorem 18] W [ ] -hard [Theorem 16] W [ ] -hard [Theorem 19] k a W [ ] -hard [Theorem 22] W [ ] -hard [Theorem 23] W [ ] -hard [Theorem 25] W [ ] -hard [Theorem 26] ( k a , k d ) O ∗ ( k k d a ) [Theorem 28]No poly kernel [Theorem 27] m para- NP -hard [Theorem 13] para- coNP -hard [Theorem 13] Table 1
Summary of parameterized complexity results. k d : the maximum number of voter groupsthat the defender can defend. k a : the maximum number of voter groups that the attacker can attack. m : the number of candidates. The study of controlling elections is fundamental to computational social choice: it is widelystudied from a theoretical perspective, and has deep practical impact. Bartholdi et al [4]initiated the study of these problems from a computational perspective, hoping that compu-tational hardness of these problems may suggest a substantial barrier to the phenomena ofcontrol: if it is, say NP -hard to control an election, then the manipulative agent may notbe able to compute an optimal control strategy in a reasonable amount of time. This basicapproach has been intensely studied in various other scenarios. For instance, Faliszewskiet al. [27] studied the problem of control where different types of attacks are combined(multimode control), Mattei et al [44] showed hardness of a variant of control which justexercises different tie-breaking rules, Bulteau et al. [10] studied voter control in a combin-atorial setting, etc [49, 52, 28, 11, 43, 31, 30, 29, 26, 45, 25, 24, 24, 34, 37, 33, 36, 32, 47,48, 51, 14, 21, 20, 16, 17, 15].Exploring parameterized complexity of various control problems has also gained a lot ofinterest. For example, Betzler and Uhlmann [6] studied parameterized complexity of can-didate control in elections and showed interesting connection with digraph problems, Liuand Zhu [41, 42] studied parameterized complexity of control problem by deleting votersfor many common voting rules, and so on [40, 53, 38, 18, 22]. Studying election controlfrom a game theoretic approach using security games is also an active area of research. See,for example, the works of An et al. and Letchford et al. [3, 39].The broad theme of using computational hardness as a barrier to control has two distinct lim-itations: one is, of course, that some voting rules simply remain computationally vulnerableto many forms of control, in the sense that optimal strategies can be found in polynomialtime. The other is that even NP -hard control problems often admit reasonable heuristics,can be approximated well, or even admit efficient exact algorithms in realistic scenarios.Therefore, relying on NP -hardness alone is arguably not a robust strategy against control.To address this issue, the work of Yin et al. [56] explicitly defined the problem of protect-ing an election from control , where in addition to the manipulative agent, we also have a“defender”, who can also deploy some resources to spoil a planned attack. In this setting,elections are defined with respect to voter groups rather than voters, which is a small dif-ference from the traditional control setting. The voter groups model allows us to consider . Dey, N. Misra, S. Nath, and G. Shakya 3 attacks on sets of voters, which is a more accurate model of realistic control scenarios.In Yin et al. [56], the defense problem is modeled as a Stackelberg game in which limitedprotection resources (say k d ) are deployed to protect a collection of voter groups and theadversary responds by attempting to subvert the election (by attacking, say, at most k a groups). They consider the plurality voting rule, and show that the problem of choosing theminimal set of resources that guarantee that an election cannot be controlled is NP -hard.They further suggest a Mixed-Integer Program formulation that can usually be efficientlytackled by solvers. Our main contribution is to study this problem in a parameterized settingand provide a refined complexity landscape for it. We also introduce the complementaryattack problem, and extend the study to voting rules beyond plurality. We now turn to asummary of our contributions. Contribution:
We refer the reader to Section 2 for the relevant formal definitions, while focusing here on ahigh-level overview of our results. Recall that the O
PTIMAL D EFENSE problem asks for a setof at most k d voter groups which, when protected, render any attack on at most k a votergroups unsuccessful. In this paper, we study the parameterized complexity of O PTIMAL D E - FENSE for all scoring rules and the Condorcet voting rule (these are natural choices becausethey are computationally vulnerable to control - - the underlying “attack problem” can beresolved in polynomial time). We show that the problem of finding an optimal defense istractable when both the attacker and the defender have limited resources. Specifically, weshow that the problem is fixed-parameter tractable with the combined parameter ( k a , k d ) by a natural bounded-depth search tree approach. We also show that the O PTIMAL D EFENSE problem is unlikely to admit a polynomial kernel under plausible complexity theoretic as-sumption. We observe that both these parameters are needed for fixed parameter tractab-ility, as we show W [ ] -hardness when O PTIMAL D EFENSE is parameterized by either k a or k d .Another popular parameter considered for voting problems is m , the number of candidates— as this is usually small compared to the size of the election in traditional applicationscenarios. Unfortunately, we show that O PTIMAL D EFENSE is NP -hard even when the electionhas only 3 candidates, eliminating the possibility of fixed-parameter algorithms (and evenXP algorithms). This strengthens a hardness result shown in Yin et al. [56]. Our hardnessresults on a constant number of candidates rely on a succinct encoding of the informationabout the scores of the candidates from each voter group. We also observe that the problemis polynomially solvable when only two candidates are involved.We introduce the complementary problem of attacking an election: here the attacker playsher strategy first, and the defender is free to defend any of the attacked groups within thebudget. The attacker wins if she is successful in subverting the election no matter whichdefense is played out. This problem turns out to be harder: it is already W [ ] -hard whenparameterized by both k a and k d , which is in sharp contrast to the O PTIMAL D EFENSE prob-lem. This problem is also hard in the setting of a constant number of candidates — specific-ally, it is coNP -hard for the plurality voting rule [Theorem 10] and the Condorcet votingrule [Theorem 12] even when we have only three candidates if every voter group is en-coded as the number of plurality votes every candidate receives from that voter group. Ourdemonstration of the hardness of the attack problem is another step in the program of using
A Parameterized Perspective on Protecting Elections computational intractability as a barrier to undesirable phenomenon, which, in this context,is the act of planning a systematic attack on voter groups with limited resources.We finally propose two simple greedy algorithms for the O
PTIMAL D EFENSE problem andempirically show that it may be able to solve many instances of practical interest.
Let C = { c , c , . . . , c m } be a set of candidates and V = { v , v , . . . , v n } a set of voters. Ifnot mentioned otherwise, we denote the set of candidates by C , the set of voters by V , thenumber of candidates by m , and the number of voters by n . Every voter v i has a preferenceor vote ≻ i which is a complete order over C . We denote the set of all complete orders over C by L ( C ) . We call a tuple of n preferences ( ≻ , ≻ , · · · , ≻ n ) ∈ L ( C ) n an n -voter preferenceprofile. Often it is convenient to view a preference profile as a multi-set consisting of itsvotes. The view we are taking will be clear from the context. A voting rule (often calledvoting correspondence) is a function r : ∪ n ∈ N L ( C ) n −→ C \ { ∅ } which selects, from apreference profile, a nonempty set of candidates as the winners. We refer the reader to [9]for a comprehensive introduction to computational social choice. In this paper we will befocusing on two voting rules – the scoring rules and the Condorcet voting rule which aredefined as follows. Scoring Rule:
A collection of m -dimensional vectors −→ s m = ( α , α , . . . , α m ) ∈ R m with α > α > . . . > α m and α > α m for every m ∈ N naturally defines a voting rule —a candidate gets score α i from a vote if it is placed at the i th position, and the score ofa candidate is the sum of the scores it receives from all the votes. The winners are thecandidates with the highest score. Given a set of candidates C , a score vector −→ α of length |C| , a candidate x ∈ C , and a profile P , we denote the score of x in P by s −→ α P ( x ) . Whenthe score vector −→ α is clear from the context, we omit −→ α from the superscript. A straightforward observation is that the scoring rules remain unchanged if we multiply every α i byany constant λ > µ . Hence, we assume without loss of generalitythat for any score vector −→ s m , there exists a j such that α j − α j + = α k = k > j .We call such a score vector a normalized score vector . Weighted Majority Graph and Condorcet Voting Rule:
Given an election E = ( C , ( ≻ , ≻ , . . . , ≻ n )) and two candidates x , y ∈ C , let us define N E ( x , y ) to be the number of voteswhere the candidate x is preferred over y . We say that a candidate x defeats another can-didate y in pairwise election if N E ( x , y ) > N E ( y , x ) . Using the election E , we can constructa weighted directed graph G E = ( U = C , E ) as follows. The vertex set U of the graph G E is the set of candidates C . For any two candidates x , y ∈ C with x = y , let us define themargin D E ( x , y ) of x from y to be N E ( x , y ) − N E ( y , x ) . We have an edge from x to y in G E if D E ( x , y ) >
0. Moreover, in that case, the weight w ( x , y ) of the edge from x to y is D E ( x , y ) .A candidate c is called the Condorcet winner of an election E if there is an edge from c toevery other vertices in the weighted majority graph G E . The Condorcet voting rule outputsthe Condorcet winner if it exists and outputs the set C of all candidates otherwise.Let r be a voting rule. We study the r -O PTIMAL D EFENSE problem which was defined byYin et al. [56]. It is defined as follows. Intuitively, the r -O PTIMAL D EFENSE problem asks ifthere is a way to defend k d voter groups such that, irrespective of which k a voter groups the . Dey, N. Misra, S. Nath, and G. Shakya 5 attacker attacks, the output of the election (that is the winning set of candidates) is alwayssame as the original one. A voter group gets deleted if only if it is attacked but not defended. ◮ Definition 1 ( r - O PTIMAL D EFENSE ) . Given n voter groups G i , i ∈ [ n ] , two integers k a and k d , does there exist an index set I ⊆ [ n ] with |I| k d such that, for every I ′ ⊂ [ n ] \ I with |I ′ | k a , we have r (( G i ) i ∈ [ n ] \I ′ ) = r (( G i ) i ∈ [ n ] ) ? The integers k a and k d are called respectivelyattacker’s resource and defender’s resource. We denote an arbitrary instance of the r - O PTIMAL D EFENSE problem by ( C , {G i : i ∈ [ n ] } , k a , k d ) . We also study the r -O PTIMAL A TTACK problem which is defined as follows. Intuitively, in the r -O PTIMAL A TTACK problem the attacker is interested to know if it is possible to attack k a voter groups such that, no matter which k d voter groups the defender defends, the outcomeof the election is never same as the original (that is the attack is successful). ◮ Definition 2 ( r - O PTIMAL A TTACK ) . Given n voter groups G i , i ∈ [ n ] , two integers k a and k d ,does there exist an index set I ⊆ [ n ] with |I| k a such that, for every I ′ ⊆ [ n ] with |I ′ | k d ,we have r (( G i ) i ∈ [ n ] \ ( I\I ′ ) ) = r (( G i ) i ∈ [ n ] ) ? We denote an arbitrary instance of the r - O PTIMAL A TTACK problem by ( C , {G i : i ∈ [ n ] } , k a , k d ) . Encoding of the Input Instance:
In both the r -O PTIMAL D EFENSE and r -O PTIMAL A TTACK problems, we assume that every input voter group G is encoded as follows. The encodinglists all the different votes ≻ that appear in the voter group G along with the number oftimes the vote ≻ appear in G . Hence, if a voter group G contains only k different votes over m candidates and consists of n voters, then the encoding of G takes O ( km log m log n ) bitsof memory. Parameterized complexity:
In parameterized complexity, each problem instance comeswith a parameter k . Formally, a parameterized problem Π is a subset of Γ ∗ × N , where Γ isa finite alphabet. An instance of a parameterized problem is a tuple ( x , k ) , where k is theparameter. A central notion is fixed parameter tractability (FPT) which means, for a giveninstance ( x , k ) , solvability in time f ( k ) · p ( | x | ) , where f is an arbitrary function of k and p isa polynomial in the input size | x | . Just as NP-hardness is used as evidence that a problemprobably is not polynomial time solvable, there exists a hierarchy of complexity classes aboveFPT, and showing that a parameterized problem is hard for one of these classes is consideredevidence that the problem is unlikely to be fixed-parameter tractable. The main classes inthis hierarchy are: FPT ⊆ W[1] ⊆ W[2] ⊆ · · · ⊆
W[P] ⊆ XP. We now define the notion ofparameterized reduction [13]. ◮ Definition 3.
Let A , B be parameterized problems. We say that A is fpt-reducible to B ifthere exist functions f , g : N → N , a constant α ∈ N and an algorithm Φ which transforms aninstance ( x , k ) of A into an instance ( x ′ , g ( k )) of B in time f ( k ) | x | α so that ( x , k ) ∈ A if andonly if ( x ′ , g ( k )) ∈ B . To show W-hardness in the parameterized setting, it is enough to give a parameterizedreduction from a known hard problem. For a more detailed and formal introduction toparameterized complexity, we refer the reader to [13] for a detailed introduction to thisparadigm. ◮ Definition 4. [Kernelization] [50, 35] A kernelization algorithm for a parameterized prob-lem Π ⊆ Γ ∗ × N is an algorithm that, given ( x , k ) ∈ Γ ∗ × N , outputs, in time polynomial in | x | + k , a pair ( x ′ , k ′ ) ∈ Γ ∗ × N such that (a) ( x , k ) ∈ Π if and only if ( x ′ , k ′ ) ∈ Π and (b) | x ′ | , k ′ g ( k ) , where g is some computable function. The output instance x ′ is called the kernel, A Parameterized Perspective on Protecting Elections and the function g is referred to as the size of the kernel. If g ( k ) = k O ( ) , then we say that Π admits a polynomial kernel. For many parameterized problems, it is well established that the existence of a polynomialkernel would imply the collapse of the polynomial hierarchy to the third level (or moreprecisely,
CoNP ⊆ NP / Poly ). Therefore, it is considered unlikely that these problems wouldadmit polynomial-sized kernels. For showing kernel lower bounds, we simply establishreductions from these problems. ◮ Definition 5. [Polynomial Parameter Transformation] [8]
Let Γ and Γ be paramet-erized problems. We say that Γ is polynomial time and parameter reducible to Γ , written Γ Ptp Γ , if there exists a polynomial time computable function f : Σ ∗ × N → Σ ∗ × N , and apolynomial p : N → N , and for all x ∈ Σ ∗ and k ∈ N , if f (( x , k )) = ( x ′ , k ′ ) , then ( x , k ) ∈ Γ ifand only if ( x ′ , k ′ ) ∈ Γ , and k ′ p ( k ) . We call f a polynomial parameter transformation (ora PPT) from Γ to Γ . This notion of a reduction is useful in showing kernel lower bounds because of the followingtheorem. ◮ Theorem 6. [8, Theorem 3]
Let P and Q be parameterized problems whose derived classicalproblems are P c , Q c , respectively. Let P c be NP − complete , and Q c ∈ NP . Suppose there existsa PPT from P to Q . Then, if Q has a polynomial kernel, then P also has a polynomial kernel. Yin et al. [56] showed that the O
PTIMAL D EFENSE problem is polynomial time solvable forthe plurality voting rule when we have only 2 candidates. On the other hand, they alsoshowed that the O
PTIMAL D EFENSE problem is NP -complete when we have an unbounded number of candidates. We begin with improving their NP -completeness result by showingthat the O PTIMAL D EFENSE problem becomes NP -complete even when we have only 3 can-didates and the attacker can attack any number of voter groups. Towards that, we reducethe k -S UM problem to the O PTIMAL D EFENSE problem. The k -S UM problem is defined asfollows. ◮ Definition 7 ( k - S UM ) . Given a set of n positive integers W = { w i , i ∈ [ n ] } , and two positiveintegers k n and M , does there exist an index set I ⊂ [ n ] with |I| = k such that P i ∈ I w i = M ? The k -S UM problem can be easily proved to be NP -complete by modifying the NP -completeness proof of the Subset Sum problem in Cormen et al. [12]. We also need thefollowing structural result for normalized scoring rules which has been used before [5, 19]. ◮ Lemma 8.
Let C = { c , . . . , c m } be a set of candidates and −→ α a normalized score vector oflength |C| . Let x , y ∈ C , x = y , be any two arbitrary candidates. Then there exists a profile P yx consisting of m votes such that we have the following. s P yx ( x ) + = s P yx ( y ) − = s P yx ( a ) for every a ∈ C \ { x , y } For any two candidates x , y ∈ C , x = y , we use P yx to denote the profile as defined inTheorem 8. We are now ready to present our NP -completeness result for the O PTIMAL D EFENSE problem for the scoring rules even in the presence of 3 candidates only. In theinterest of space, we will provide only a sketch of a proof for a several results. . Dey, N. Misra, S. Nath, and G. Shakya 7 ◮ Theorem 9.
The O PTIMAL D EFENSE problem is NP -complete for every scoring rule even ifthe number of candidates is and the attacker can attack any number of the voter groups. Proof.
The O
PTIMAL D EFENSE problem for every scoring rule can be shown to belong to NP by using a defense strategy S (a subset of at most k d voter groups) as a certificate. The factthat the certificate can be validated in polynomial time involves checking if there exists asuccessful attack despite protecting all groups in S . This can be done in polynomial time, butdue to space constraints, we defer a detailed argument to a full version of this manuscript.We now turn to the reduction from k -S UM .Let −→ α be any normalized score vector of length 3. The O PTIMAL D EFENSE problem forthe scoring rule based on −→ α belongs to NP . Let ( W = { w , . . . , w n } , k , M ) be an arbitraryinstance of the k -S UM problem. We can assume, without loss of generality, that 8 divides M and w i for every i ∈ [ n ] ; if this is not the case, we replace M and w i by respectively 8 M and8 w i for every i ∈ [ n ] which clearly is an equivalent instance of the original instance. Let usalso assume, without loss of generality, that 2 k < n (if not then add enough copies of M + W ) and M < P ni = w i (since otherwise, it is a trivial N O instance). We construct thefollowing instance of the O PTIMAL D EFENSE problem for the scoring rule based on −→ α . Let M ′ be an integer such that M ′ > P ni = w i and 8 divides M ′ . We have 3 candidates, namely a , b , and c . We have the following voter groups.– For every i ∈ [ n ] , we have a voter group G i consisting of w i copies of P ca (as defined inTheorem 8) and M ′ − w i copies of P cb . Hence, we have the following. s G i ( c ) = s G i ( a ) + M ′ + w i = s G i ( b ) + M ′ − w i – We have one voter group ˆ G consisting of ( kM ′ + M ) / − P ac , ( kM ′ − M ) / − P bc , and ( kM ′ − M ) / − P ba . We have the following. s ˆ G ( c ) = s ˆ G ( a ) − ( kM ′ + M − ) = s ˆ G ( b ) − ( kM ′ − M − ) Let Q be the resulting profile; that is Q = ∪ ni = G i ∪ ˆ G . We have s Q ( c ) = s Q ( a ) + ( n − k ) M ′ + P ni = w i − M + = s Q ( b )+( n − k ) M ′ + M − P ni = w i +
6. Since n > k and M ′ > P ni = w i ,we have s Q ( c ) > s Q ( a ) and s Q ( c ) > s Q ( b ) . Thus the candidate c wins the election uniquely.We define k d , the maximum number of voter groups that the defender can defend, to be k . We define k a , the maximum number of voter groups that the attacker can attack, to be n +
1. This finishes the description of the O
PTIMAL D EFENSE instance. We claim that thetwo instances are equivalent.In the forward direction, let the k -S UM instance be a Y ES instance and I ⊂ [ n ] with |I| = k be an index set such that P i ∈ I w i = M . Let us consider the defense strategy wherethe defender protects the voter groups G i for every i ∈ I . Since P i ∈ I w i = M , we have P i ∈ I ( M ′ − w i ) = kM ′ − M . Let H be the profile of voter groups corresponding to the indexset I ; that is, H = ∪ i ∈ I G i . Let H ′ be the profile remaining after the attacker attacks somevoter groups. Without loss of generality, we can assume that the attacker does not attack thevoter group ˆ G since otherwise the candidate c continues to win uniquely. We thus obviouslyhave H ∪ ˆ G ⊆ H ′ . We have s H ∪ ˆ G ( c ) = s H ∪ ˆ G ( a ) + kM ′ + P i ∈ I w i − ( kM ′ + M − ) = s H ∪ ˆ G ( a ) + s H ∪ ˆ G ( c ) = s H ∪ ˆ G ( b ) + kM ′ − P i ∈ I w i − ( kM ′ − M − ) = s H ∪ ˆ G ( b ) + c receives as much score as any other candidate in the voter group G i for every i ∈ [ n ] , we have s H ′ ∪ ˆ G ( c ) > s H ′ ∪ ˆ G ( a )+ s H ′ ∪ ˆ G ( c ) > s H ′ ∪ ˆ G ( b )+
6. Hence, thecandidate c wins uniquely in the resulting profile H ′ after the attack and thus the defense issuccessful. A Parameterized Perspective on Protecting Elections
In the other direction, let the O
PTIMAL D EFENSE instance be a Y ES instance. Without loss ofgenerality, we can assume that the attacker does not attack the voter group ˆ G and thus thedefender does not defend the voter group ˆ G . We can also assume, without loss of generality,that the defender defends exactly k voter groups since the candidate c receives as much scoreas any other candidate in the voter group G i for every i ∈ [ n ] . Let I ⊂ [ n ] with |I| = k suchthat defending all the voter groups G i , i ∈ I is a successful defense strategy. We claim that P i ∈ I w i > M . Suppose not, then let us assume that P i ∈ I w i < M . Since, w i is divisibleby 8 and positive for every i ∈ [ n ] and m is divisible by 8, we have P i ∈ I w i M −
8. Let H be the profile of voter groups corresponding to the index set I ; that is, H = ∪ i ∈ I G i . Wehave s H ∪ ˆ G ( c ) = s H ∪ ˆ G ( a ) + kM ′ + P i ∈ I w i − ( kM ′ + M − ) s H ∪ ˆ G ( a ) + M − − M + = s H ∪ ˆ G ( a ) −
2. Hence attacking the voter groups G i , i ∈ [ n ] \ I makes the score of c strictlyless than the score of a . This contradicts our assumption that defending all the voter groups G i , i ∈ I is a successful defense strategy. Hence we have P i ∈ I w i > M . We now claim that P i ∈ I w i M . Suppose not, then let us assume that P i ∈ I w i > M . Since, w i is divisible by8 and positive for every i ∈ [ n ] and m is divisible by 8, we have P i ∈ I w i > M +
8. Let H ′ bethe profile of voter groups corresponding to the index set I ; that is, H ′ = ∪ i ∈ I G i . We have s H ′ ∪ ˆ G ( c ) = s H ′ ∪ ˆ G ( b ) + kM ′ − P i ∈ I w i − ( kM ′ − M − ) s H ′ ∪ ˆ G ( b ) − ( M + ) + M + = s H ′ ∪ ˆ G ( b ) −
2. Hence attacking the voter groups G i , i ∈ [ n ] \ I makes the score of c strictlyless than the score of b . This contradicts our assumption that defending all the voter groups G i , i ∈ I is a successful defense strategy. Hence we have P i ∈ I w i M . Therefore we have P i ∈ I w i = M and thus the k -S UM instance is a Y ES instance. ◭ In the proof of Theorem 9, we observe that the reduced instance of the O
PTIMAL D EFENSE problem viewed as an instance of the O
PTIMAL A TTACK problem is a N O instance if andonly if the k -S UM instance is a Y ES instance. Hence, the same reduction as in the proof ofTheorem 9 gives us the following result for the O PTIMAL A TTACK problem. ◮ Corollary 10.
The O PTIMAL A TTACK problem is coNP -hard for every scoring rule even if thenumber of candidates is and the attacker can attack any number of voter groups. We now prove a similar hardness result as of Theorem 9 for the Condorcet voting rule. ◮ Theorem 11.
The O PTIMAL D EFENSE problem is NP -complete for the Condorcet voting ruleeven if the number of candidates is and the attacker can attack any number of voter groups. Proof.
The O
PTIMAL D EFENSE problem for the Condorcet voting rule clearly belongs to NP . To show NP -hardness, we reduce an arbitrary instance of the k -S UM problem to theO PTIMAL D EFENSE problem for the Condorcet voting rule. Let ( { w , . . . , w n } , k , M ) be anarbitrary instance of the k -S UM problem. We construct the following instance of the O PTIMAL D EFENSE problem for the Condorcet voting rule. Let M ′ = max { w i : i ∈ [ n ] } . We have 3candidates, namely a , b , and c . We have the following voter groups.– For every i ∈ [ n ] , we have a voter group G i where D G i ( a , b ) = w i , D G i ( a , c ) = ( M ′ − w i ) , and D G i ( b , c ) = G where the candidates b and c receive respectively D ˆ G ( b , a ) = M − D ˆ G ( c , a ) = ( kM ′ − M ) −
1, and D ˆ G ( b , c ) = k d , the maximum number of voter groups that the defender can defend, to be k . We define k a , the maximum number of voter groups that the attacker can attack, tobe n +
1. We observe that the candidate a is the Condorcet winner of the election. This . Dey, N. Misra, S. Nath, and G. Shakya 9 finishes the description of the O PTIMAL D EFENSE instance. We claim that the two instancesare equivalent.In the forward direction, let the k -S UM instance be a Y ES instance and I ⊂ [ n ] with |I| = k be an index set such that P i ∈ I w i = M . Let us consider the defense strategy wherethe defender protects the voter groups G i for every i ∈ I . Since P i ∈ I w i = M , we have P i ∈ I ( M ′ − w i ) = kM ′ − M . Without loss of generality, we can assume that the attackerdoes not attack the voter group ˆ G . We observe that the candidate a is the Condorcet winnerof the election even when the attacker attacks all the voter groups G j , j ∈ [ n ] \ I . Hence theO PTIMAL D EFENSE instance is a Y ES instance.In the other direction, let the O PTIMAL D EFENSE instance be a Y ES instance. Without loss ofgenerality, we can assume that the attacker does not attack the voter group ˆ G and thus thedefender does not defend the voter group ˆ G . We can also assume, without loss of generality,that the defender defends exactly k voter groups since the candidate a continues to be theCondorcet winner if the attacker attacks at most k − I ⊂ [ n ] with |I| = k such that defending all the voter groups G i , i ∈ I is a successful defense strategy.We claim that P i ∈ I w i > M . Suppose not, then let us assume that P i ∈ I w i < M . Thenattacking the voter groups G i , i ∈ [ n ] \ I makes the candidate b defeat the candidate a in pairwise election. This contradicts or assumption that defending all the voter groups G i , i ∈ I is a successful defense strategy. Hence we have P i ∈ I w i > M . We now claim that P i ∈ I w i M . Suppose not, then let us assume that P i ∈ I w i > M . Then attacking thevoter groups G i , i ∈ [ n ] \ I makes the candidate c defeat the candidate a in pairwise election.This contradicts or assumption that defending all the voter groups G i , i ∈ I is a successfuldefense strategy. Hence we have P i ∈ I w i M . Therefore we have P i ∈ I w i = M and thusthe k -S UM instance is a Y ES instance. ◭ In the proof of Theorem 11, we observe that the reduced instance of O
PTIMAL D EFENSE viewed as an instance of the O
PTIMAL A TTACK problem is a N O instance if and only if the k -S UM instance is a Y ES instance. Hence, the same reduction as in the proof of Theorem 11gives us the following result for the O PTIMAL A TTACK problem. ◮ Corollary 12.
The O PTIMAL A TTACK problem is coNP -hard for the Condorcet voting ruleeven if the number of candidates is and the attacker can attack any number of voter groups. In this section, we present our hardness results for the O
PTIMAL D EFENSE and the O
PTIMAL A TTACK problems in the parameterized complexity framework. We consider the followingparameters for both the problems – number of candidate ( m ), defender’s resource ( k d ), andattacker’s resource ( k a ). From Theorems 9 to 12 we immediately have the following resultfor the O PTIMAL D EFENSE and O
PTIMAL A TTACK problems parameterized by the number ofcandidates for both the scoring rules and the Condorcet voting rule. ◮ Corollary 13.
The O PTIMAL D EFENSE problem is para- NP -hard parameterized by the num-ber of candidates for both the scoring rules and the Condorcet voting rule. The O PTIMAL A TTACK problem is para- coNP -hard parameterized by the number of candidates for both the scoringrules and the Condorcet voting rule.
The NP -completeness proof for the O PTIMAL D EFENSE problem for the plurality voting ruleby Yin et al. [56] is actually a parameter preserving reduction from the H
ITTING S ET problemparameterized by the solution size. The H ITTING S ET problem is defined as follows. ◮ Definition 14 ( H ITTING S ET ) . Given a universe U , a set S = { S i : i ∈ [ t ] } of subsets of U ,and a positive integer k which is at most | U | , does there exist a subset W ⊆ U with |W| = k suchthat W ∩ S i = ∅ for every i ∈ [ t ] . We denote an arbitrary instance of H ITTING S ET by ( U , S , k ) . Since the H
ITTING S ET problem parameterized by the solution size k is known to be W [ ] -complete [23], the following result immediately follows from Theorem 2 of Yin et al. [56]. ⊲ Observation 1 ([56]).
The O
PTIMAL D EFENSE problem for the plurality voting rule is W [ ] -hard parameterized by k d .We now generalize Observation 1 to any scoring rule by exhibiting a polynomial parametertransform from the H ITTING S ET problem parameterized by the solution size. ◮ Theorem 15.
The O PTIMAL D EFENSE problem for every scoring rule is W [ ] -hard paramet-erized by k d . Proof.
Let ( U , S = { S j : j ∈ [ t ] } , k ) be an arbitrary instance of H ITTING S ET . Let U = { z i : i ∈ [ n ] } . Without loss of generality, we assume that S j = ∅ for every j ∈ [ t ] since otherwise theinstance is a N O instance. Let −→ α be a normalized score vector of length t +
2. We constructthe following instance of the O
PTIMAL D EFENSE problem for the scoring rule based on −→ α .The set of candidates C = { x j : j ∈ [ t ] } ∪ { y , d } . We have the following voter groups.– For every i ∈ [ n ] , we have a voter group G i . For every j ∈ [ t ] with z i ∈ S j we have 2copies of P dx j in G i .– We have one group ˆ G where we have 2 tn copies of P x j d for every j ∈ [ n ] and 2 tn − P yd .Let Q be the resulting profile; that is Q = ∪ ni = G i ∪ ˆ G . We define the defender’s resource k d to be k + n . This finishes the description of the O PTIMAL D EFENSE instance. Since S j = ∅ for every j ∈ [ t ] , we have s Q ( y ) > s Q ( x j ) for every j ∈ [ t ] .We also have s Q ( y ) > s Q ( d ) . Hence the candidate y is the unique winner of the profile Q . Wenow prove that the O PTIMAL D EFENSE instance ( C , Q , k a , k d ) is equivalent to the H ITTING S ET instance ( U , S , k ) .In the forward direction, let us suppose that the H ITTING S ET instance is a Y ES instance. Let I ⊂ [ n ] be such that |I| = k and { z i : i ∈ I} ∩ S j = ∅ . We claim that the defender’s strategy ofdefending the voter groups G j for every j ∈ [ t ] \ I and ˆ G results in a successful defense. Let H be the profile of voter groups corresponding to the index set I ; that is, H = ∪ i ∈ I G i . Let H ′ bethe profile remaining after the attacker attacks some voter groups. We thus obviously have H ∪ ˆ G ⊆ H ′ . Since { z i : i ∈ I} forms a hitting set, we have s H ′ ( y ) > s H ′ ( x j ) for every j ∈ [ t ] .Also since the voter group ˆ G is defended, we have s H ′ ( y ) > s H ′ ( d ) . Hence the candidate y continues to win uniquely even after the attack and hence the O PTIMAL D EFENSE instance isa Y ES instance.In the other direction, let the O PTIMAL D EFENSE instance be a Y ES instance. Without lossof generality, we can assume that the defender defends the voter group ˆ G since otherwisethe attacker can attack the voter group ˆ G which makes the score of the candidate d morethan the score of the candidate y and thus defense would fail. We can also assume, without . Dey, N. Misra, S. Nath, and G. Shakya 11 loss of generality, that the defender defends exactly k voter groups. Let I ⊂ [ n ] with |I| = k such that defending all the voter groups G i , i ∈ I and ˆ G is a successful defense strategy.Let us consider Z = { z i : i ∈ I} ⊆ U . We claim that Z must form a hitting set. Indeed,otherwise let us assume that there exists a j ∈ [ t ] such that Z ∩ S j = ∅ . Consider thesituation where the attacker attacks voter groups G i for every i ∈ [ n ] \ I . We observe that s ∪ i ∈ I G i ∪ ˆ G ( x j ) > s ∪ i ∈ I G i ∪ ˆ G ( y ) . This contradicts our assumption that defending all the votergroups G i , i ∈ I and ˆ G is a successful defense strategy. Hence Z forms a hitting set and thusthe H ITTING S ET instance is a Y ES instance. ◭ In the proof of Theorem 15, we observe that the reduced instance of O
PTIMAL D EFENSE viewed as an instance of the O
PTIMAL A TTACK problem is a N O instance if and only if the k -S UM instance is a Y ES instance. Hence, the same reduction as in the proof of Theorem 15gives us the following result for the O PTIMAL A TTACK problem. ◮ Corollary 16.
The O PTIMAL A TTACK problem for every scoring rule is W [ ] -hard parameter-ized by k d . We now show W [ ] -hardness of the O PTIMAL D EFENSE problem for the Condorcet votingrule parameterized by k d . Towards that, we need the following lemma which has been usedbefore [46, 55]. ◮ Lemma 17.
For any function f : C × C −→ Z , such that ∀ a , b ∈ C , f ( a , b ) = − f ( b , a ) . ∀ a , b , c , d ∈ C , f ( a , b ) + f ( c , d ) is even,there exists a n voters’ profile such that for all a , b ∈ C , a defeats b with a margin of f ( a , b ) .Moreover, n is even and n = O X { a , b } ∈ C × C | f ( a , b ) | Next, we show the W [ ] -hardness of the O PTIMAL D EFENSE problem for the Condorcet votingrule parameterized by k d . This is also a parameter-preserving reduction from the H ITTING S ET problem. ◮ Theorem 18.
The O PTIMAL D EFENSE problem for the Condorcet voting rule is W [ ] -hardparameterized by k d . Proof.
Let ( U , S = { S j : j ∈ [ t ] } , k ) be an arbitrary instance of H ITTING S ET . Let U = { z i : i ∈ [ n ] } . Without loss of generality, we assume that S j = ∅ for every j ∈ [ t ] since otherwisethe instance is a N O instance. We construct the following instance of the O PTIMAL D EFENSE problem for the Condorcet voting rule. The set of candidates C = { x j : j ∈ [ t ] } ∪ { y } . For every i ∈ [ n ] , we have a voter group G i . For every j ∈ [ t ] with z i ∈ S j we have D G i ( y , x j ) = Q be the resulting profile; that is Q = ∪ ni = G i . We define the defender’s resource k d tobe k and attacker’s resource to be n . This finishes the description of the O PTIMAL D EFENSE instance. Since S j = ∅ for every j ∈ [ t ] , we have D Q ( y , x j ) > j ∈ [ t ] . Hencethe candidate y is the Condorcet winner of the profile Q . We now prove that the O PTIMAL D EFENSE instance ( C , Q , k a , k d ) is equivalent to the H ITTING S ET instance ( U , S , k ) .In the forward direction, let us suppose that the H ITTING S ET instance is a Y ES instance. Let I ⊂ [ n ] be such that |I| = k and { z i : i ∈ I} ∩ S j = ∅ . We claim that the defender’s strategy of defending the voter groups G j for every j ∈ [ t ] \ I results in a successful defense. Let H be the profile of voter groups corresponding to the index set I ; that is, H = ∪ i ∈ I G i . Let H ′ be the profile remaining after the attacker attacks some voter groups. We thus obviouslyhave H ⊆ H ′ . Since { z i : i ∈ I} forms a hitting set, we have D H ′ ( y , x j ) > j ∈ [ t ] . Hence the candidate y continues to win uniquely even after the attack and hence theO PTIMAL D EFENSE instance is a Y ES instance.In the other direction, let the O PTIMAL D EFENSE instance be a Y ES instance. We can alsoassume, without loss of generality, that the defender defends exactly k voter groups. Let I ⊂ [ n ] with |I| = k such that defending all the voter groups G i , i ∈ I is a successful defensestrategy. Let us consider Z = { z i : i ∈ I} ⊆ U . We claim that Z must form a hitting set.Indeed, otherwise let us assume that there exists a j ∈ [ t ] such that Z ∩ S j = ∅ . Consider thesituation where the attacker attacks voter groups G i for every i ∈ [ n ] \ I . We observe that D ∪ i ∈ I G i ( y , x j ) = y is not the Condorcet winner. This contradictsour assumption that defending all the voter groups G i , i ∈ I is a successful defense strategy.Hence Z forms a hitting set and thus the H ITTING S ET instance is a Y ES instance. ◭ In the proof of Theorem 18, we observe that the reduced instance of O
PTIMAL D EFENSE viewed as an instance of the O
PTIMAL A TTACK problem is a N O instance if and only if the k -S UM instance is a Y ES instance. Hence, the same reduction as in the proof of Theorem 18gives us the following result for the O PTIMAL A TTACK problem. ◮ Corollary 19.
The O PTIMAL A TTACK problem for the Condorcet voting rule is W [ ] -hardparameterized by k d . We now show that the O
PTIMAL D EFENSE problem for scoring rules is W [ ] -hard paramet-erized by k a also by exhibiting a parameter preserving reduction from a problem closelyrelated to H ITTING S ET , which is S ET C OVER problem parameterized by the solution size.The S ET C OVER problem is defined as follows. This is a W [ ] -complete problem [23]. Wenow present our W [ ] -hardness proof for the O PTIMAL D EFENSE problem for scoring rulesparameterized by k a , by a reduction from S ET C OVER . ◮ Definition 20 ( S ET C OVER ) . Given an universe U , a set S = { S i : i ∈ [ t ] } of subsets of U , anda non-negative integer k which is at most t , does there exists an index set I ⊂ [ t ] with |I| = k such that S i ∈ I S i = U . We denote an arbitrary instance of S ET C OVER by ( U , S , k ) . ◮ Theorem 21.
The O PTIMAL D EFENSE problem for every scoring rule and Condorcet rule is W [ ] -hard parameterized by k a . ◮ Theorem 22.
The O PTIMAL D EFENSE problem for every scoring rule is W [ ] -hard paramet-erized by k a . Proof.
Let ( U , S = { S j : j ∈ [ t ] } , k ) be an arbitrary instance of S ET C OVER . Let U = { z i : i ∈ [ n ] } . We assume that k > ET C OVER instance is polynomialtime solvable. For i ∈ [ n ] , let f i be the number of j ∈ [ t ] such that z i ∈ S j ; that is, f i = |{ j ∈ [ t ] : z i ∈ S j }| . We assume, without loss of generality, that for every i ∈ [ n ] , t − f i − k > k by adding at most 9 t empty sets in S . We construct the following instance of theO PTIMAL D EFENSE problem for the scoring rule induced by the score vector −→ α rule. The setof candidates C = { x i : i ∈ [ n ] } ∪ { y , d } . Let −→ α be any normalized score vector of length n + . Dey, N. Misra, S. Nath, and G. Shakya 13 – For every j ∈ [ t ] , we have a voter group G j . For every i ∈ [ n ] and j ∈ [ t ] with z i / ∈ S j , wehave 2 copies of P dx i .– We have another voter group H where, for every i ∈ [ n ] , we have 2 tn + ( ( t − f i − k ) + ) copies of P x i d and 2 tn copies of P yd .We define attacker resource k a to be k and the defender’s resource k d to be t − k . Thisfinishes the description of the O PTIMAL D EFENSE instance. We first observe that the scoreof the candidate d is strictly less than the score of every other candidate. We now observethat the candidate y is the unique winner of the election since the score of the candidate y is 2 k − x i for every i ∈ [ n ] . We now prove that theO PTIMAL D EFENSE instance ( C , ∪ j ∈ [ t ] G j ∪ H , k a , k d ) is equivalent to the S ET C OVER instance ( U , S , k ) .In the forward direction, let us suppose that the S ET C OVER instance is a Y ES instance. Let I ⊂ [ t ] be such that |I| = k and S j ∈ I S j = U . We claim that the defender’s strategy ofdefending the voter groups G j for every j ∈ [ t ] \ I results in a successful defense. To seethis, we first observe that, if the attacker attacks the voter group H , then the candidate y continues to uniquely win the election irrespective of what other voter groups the attackerattacks. Indeed, since t − f i − k > k for every i ∈ [ n ] , the score of the candidate x i is strictlyless than the score of the candidate y irrespective of what other voter groups the attackerattacks. Since, for every i ∈ [ n ] and j ∈ [ t ] , the score of the candidate x i is not more thanthe score of the candidate y in the voter group G j , we may assume that the attacker attacksthe voter group G j for every j ∈ I (since they are the only voter groups unprotected except H ). Now, since S j , j ∈ I forms a set cover of U , after deleting the voter groups G j , j ∈ I , thescore of the candidate x i increases by at most 2 ( k − ) from the original election for every i ∈ [ n ] . Hence, after deleting the voter groups G j , j ∈ I , the score of the candidate x i is stillstrictly less than the score of the candidate y . Hence the candidate y continues to win andthus the defense is successful. Hence the O PTIMAL D EFENSE instance is a Y ES instance.In the other direction, let us suppose that the O PTIMAL D EFENSE instance is a Y ES instance.We assume, without loss of generality, that the defender protects exactly t − k voter groups.We argued in the forward direction that we can assume, without loss of generality, thatthe attacker never attacks the voter group H . Hence, we can also assume, without loss ofgenerality, that the defender also does not defend the voter group H . Let I ⊂ [ t ] be suchthat |I| = k and the defender defends the voter group G j for every j ∈ [ t ] \ I . We claimthat the sets S j , j ∈ I forms a set cover of U . Suppose not, then let z i be an element in U which is not covered by S j , j ∈ I . We observe that attacking the voter groups G j for every j ∈ I increases the score of the candidate x i by 2 k which makes the candidate y lose in theresulting election (after deleting the voter groups G j for every j ∈ I ) since the score of x i isstrictly more than the score of y . This contradicts our assumption that defending the votergroup G j for every j ∈ [ t ] \ I is a successful defense strategy. Hence S j , j ∈ I forms a set coverof U and thus the S ET C OVER instance is a Y ES instance. ◭ We now present our W [ ] -hardness proof for the O PTIMAL D EFENSE problem for the Con-dorcet voting rule parameterized by k a . ◮ Theorem 23.
The O PTIMAL D EFENSE problem for the Condorcet voting rule is W [ ] -hardparameterized by k a . Proof.
Let ( U , S = { S j : j ∈ [ t ] } , k ) be an arbitrary instance of S ET C OVER . Let U = { z i : i ∈ [ n ] } . We assume that k > ET C OVER instance is polynomial timesolvable. For i ∈ [ n ] , let f i be the number of j ∈ [ t ] such that z i ∈ S j ; that is, f i = |{ j ∈ [ t ] : z i ∈ S j }| . We assume, without loss of generality, that for every i ∈ [ n ] , t − f i − k > k by adding at most 9 t empty sets in S . We construct the following instance of the O PTIMAL D EFENSE problem for the Condorcet voting rule. The set of candidates C = { x i : i ∈ [ n ] } ∪ { y } .We have the following voter groups.– For every j ∈ [ t ] , we have a voter group G j . For every i ∈ [ n ] and j ∈ [ t ] , we have D G j ( y , x i ) = z i / ∈ S j and D G j ( y , x i ) = D G j ( x i , x ℓ ) = j ∈ [ t ] , i , ℓ ∈ [ n ] with i = ℓ .– We have another voter group H where, for every i ∈ [ n ] , we have D H ( x i , y ) = ( t − f i − k ) . We also have D H ( x i , x ℓ ) = i , ℓ ∈ [ n ] with i = ℓ .We define attacker resource k a to be k and the defender’s resource k d to be t − k . This fin-ishes the description of the O PTIMAL D EFENSE instance. We first observe that the candidate y is a Condorcet winner of the resulting election. We now prove that the O PTIMAL D EFENSE instance ( C , ∪ j ∈ [ t ] G j ∪ H , k a , k d ) is equivalent to the S ET C OVER instance ( U , S , k ) .In the forward direction, let us suppose that the S ET C OVER is a Y ES instance. Let I ⊂ [ t ] be such that |I| = k and S j ∈ I S j = U . We claim that the defender’s strategy of defendingthe voter groups G j for every j ∈ [ t ] \ I results in a successful defense. To see this, we firstobserve that, we can assume without loss of generality that the attacker does not attack thevoter group H since the candidate y loses every pairwise election in H . Since, for every i ∈ [ n ] and j ∈ [ t ] , the candidate y does not lose any pairwise election in the voter group G j , we may assume that the attacker attacks the voter group G j for every j ∈ I (since theyare the only voter groups unprotected except H ). Now, since S j , j ∈ I forms a set cover of U , after deleting the voter groups G j , j ∈ I , we have D ∪ j ∈ [ t ] \I G i ∪ H ( y , x i ) > ( t − f i − k + ) − ( t − f i − k ) = i ∈ [ n ] . Hence, after deleting the voter groups G j , j ∈ I ,the candidate y continues to be the Condorcet winner of the remaining profile. Hence theO PTIMAL D EFENSE instance is a Y ES instance.In the other direction, let us suppose that the O PTIMAL D EFENSE instance is a Y ES instance.We assume, without loss of generality, that the defender protects exactly t − k voter groups.We argued in the forward direction that we can assume, without loss of generality, thatthe attacker never attacks the voter group H . Hence, we can also assume, without loss ofgenerality, that the defender also does not defend the voter group H . Let I ⊂ [ t ] be such that |I| = k and the defender defends the voter group G j for every j ∈ [ t ] \ I . We claim that thesets S j , j ∈ I forms a set cover of U . Suppose not, then let z i be an element in U which is notcovered by S j , j ∈ I . We observe that D ∪ j ∈ [ t ] \I G i ∪ H ( y , x i ) = ( t − f i − k )− ( t − f i − k ) = G j for every j ∈ I makes the candidate y not the Condorcetwinner. This contradicts our assumption that defending the voter group G j for every j ∈ [ t ] \I is a successful defense strategy. Hence S j , j ∈ I forms a set cover of U and thus the S ET C OVER instance is a Y ES instance. ◭ We now show that the O
PTIMAL A TTACK problem for the scoring rules is W [ ] -hard evenparameterized by the combined parameter k a and k d . Towards that, we exhibit a polyno-mial parameter transform from the C LIQUE problem parameterized by the size of the cliquewe are looking for which is known to be W [ ] -complete. The C LIQUE problem is defined asfollows. . Dey, N. Misra, S. Nath, and G. Shakya 15 ◮ Definition 24 ( C LIQUE ) . Given a graph G and an integer k , does there exist a clique in G ofsize k ? We denote an arbitrary instance of C LIQUE by ( G , k ) . ◮ Theorem 25.
The O PTIMAL A TTACK problem for every scoring rule is W [ ] -hard parameter-ized by ( k a , k d ) . Proof.
Let ( G = ( V , E ) , k ) be an arbitrary instance of the C LIQUE problem. Let V = { v i : i ∈ [ n ] } and E = { e j : j ∈ [ m ] } . Let −→ α be any arbitrary normalized score vector of length m +
2. We construct the following instance of the O
PTIMAL A TTACK problem for the scoringrule induced by the score vector −→ α . The set of candidates C = { x j : j ∈ [ m ] } ∪ { y , d } . We havethe following voter groups.– For every i ∈ [ n ] , we have a voter group G i . For every i ∈ [ n ] , we have 10 m copies of P xd for every x ∈ C \ { d } in G i . We also have two copies of P dx j in the voter group G i if theedge e j is incident on the vertex v i , for every i ∈ [ m ] and j ∈ [ m ] .– We have another voter group H . We have one copy of P x j d for every j ∈ [ m ] in H .We define attacker resource k a to be k and the defender’s resource k d to be k −
2. Thisfinishes the description of the O
PTIMAL A TTACK instance. Let Q be the resulting profile; thatit Q = ∪ i ∈ [ n ] G i ∪ H . We first observe that the candidate y is the winner of the resultingelection since s Q ( y ) = s Q ( x j ) + s Q ( y ) > s Q ( d ) . This completes a description of theconstruction. Due to lack of space, we defer the proof of equivalence to a longer version ofthis manuscript. We now prove that the O PTIMAL A TTACK instance ( C , Q , k a , k d ) is equivalentto the C LIQUE instance ( G , k ) .In the forward direction, let us assume that U = { v i : i ∈ I} ⊂ V with |I| = k forms a cliquein G . We claim that attacking all the voter groups G i , i ∈ I forms a successful attack. Indeed,suppose the defender defends all the voter groups G i , i ∈ I except G ℓ and G ℓ ′ . Let e j ⋆ bethe edge between the vertices v ℓ and v ℓ ′ in G . Let the profile after the attack be ˆ G ; that is,ˆ G = ∪ i ∈ [ n ] \I G i ∪ G ℓ ∪ G ℓ ′ ∪ H . Then we have s ˆ G ( y ) = s ˆ G ( x j ⋆ ) − y does not win after the attack. Hence the O PTIMAL A TTACK instance is Y ES instance.In the other direction, let the O PTIMAL A TTACK instance be a Y ES instance. We first observethat the candidate d performs worse than everyone else in every voter group and thus d cannever win. Now we can assume, without loss of generality, that the attacker does not attackthe voter group H since the candidate y is not receiving more score than any other candidateexcept d in H . Let attacking all the voter groups G i , i ∈ I with |I| k is a successful attack.We observe that if |I| < k , then defending any k − y continues to win even after deleting any one group. Hence wehave |I| = k . Let us consider the subset of vertices U = { v i : i ∈ I} . We claim that U forms aclique in G . Indeed, if not, then let us assume that there exists two indices ℓ , ℓ ′ ∈ I such thatthere is no edge between the vertices v ℓ and v ℓ ′ in G . Let us consider the defender strategyof defending all the voter groups G i , i ∈ I \ { ℓ , ℓ ′ } . We observe that the candidate y continuesto uniquely receive the highest score among all the candidates and thus y wins uniquely inthe resulting election. This contradicts our assumption that attacking all the voter groups G i , i ∈ I with |I| k is a successful attack. Hence U forms a clique in G and thus the C LIQUE instance is a Y ES instance. ◭ We now show similar result as of Theorem 25 for the Condorcet voting rule. ◮ Theorem 26.
The O PTIMAL A TTACK problem for the Condorcet voting rule is W [ ] -hardparameterized by ( k a , k d ) . Proof.
Let ( G = ( V , E ) , k ) be an arbitrary instance of the C LIQUE problem. Let V = { v i : i ∈ [ n ] } and E = { e j : j ∈ [ m ] } . We construct the following instance of the O PTIMAL A TTACK problem for the Condorcet voting rule. The set of candidates C = { x j : j ∈ [ m ] } ∪ { y } . Wehave the following voter groups.– For every i ∈ [ n ] , we have a voter group G i . We have D G i ( y , x j ) = e j isincident on the vertex v i and D G i ( y , x j ) = e j is not incident on the vertex v i , for every i ∈ [ n ] and j ∈ [ m ] . We also have D G i ( x ℓ , x j ) = i ∈ [ n ] , j , ℓ ∈ [ m ] ,and j = ℓ .– We have another voter group H where we have D H ( x j , y ) = j ∈ [ m ] and D H ( x ℓ , x j ) = j , ℓ ∈ [ m ] and j = ℓ .We define attacker resource k a to be k and the defender’s resource k d to be k −
2. Thisfinishes the description of the O
PTIMAL A TTACK instance. Let Q be the resulting profile;that it Q = ∪ i ∈ [ n ] G i ∪ H . We first observe that the candidate y is the Condorcet winnerof the resulting election. We now prove that the O PTIMAL A TTACK instance ( C , Q , k a , k d ) isequivalent to the C LIQUE instance ( G , k ) .In the forward direction, let us assume that U = { v i : i ∈ I} ⊂ V with |I| = k forms a cliquein G . We claim that attacking all the voter groups G i , i ∈ I forms a successful attack. Indeed,suppose the defender defends all the voter groups G i , i ∈ I except G ℓ and G ℓ ′ . Let e j ⋆ bethe edge between the vertices v ℓ and v ℓ ′ in G . Let the profile after the attack be ˆ G ; that is,ˆ G = ∪ i ∈ [ n ] \I G i ∪ G ℓ ∪ G ℓ ′ ∪ H . Then we have D ˆ G ( y , x j ⋆ ) = y is notthe unique winner after the attack. Hence the O PTIMAL A TTACK instance is Y ES instance.In the other direction, let the O PTIMAL A TTACK instance be a Y ES instance. We can assume,without loss of generality, that the attacker does not attack the voter group H since thecandidate y loses every pairwise election in H . Let attacking all the voter groups G i , i ∈ I with |I| k is a successful attack. We observe that if |I| < k , then defending any k − y continues to be theCondorcet winner even after deleting any one group. Hence we have |I| = k . Let us considerthe subset of vertices U = { v i : i ∈ I} . We claim that U forms a clique in G . Indeed, if not,then let us assume that there exists two indices ℓ , ℓ ′ ∈ I such that there is no edge betweenthe vertices v ℓ and v ℓ ′ in G . Let us consider the defender strategy of defending all the votergroups G i , i ∈ I \ { ℓ , ℓ ′ } . We observe that the candidate y continues to be the Condorcetwinner in the resulting election. This contradicts our assumption that attacking all the votergroups G i , i ∈ I with |I| k is a successful attack. Hence U forms a clique in G and thus theC LIQUE instance is a Y ES instance. ◭ Once we have a parameterized algorithm for the O
PTIMAL D EFENSE problem for the para-meter ( k a , k d ) , an immediate question is whether there exists a kernel for the O PTIMAL D EFENSE problem of size polynomial in ( k a , k d ) . We know that the H ITTING S ET problemdoes not admit polynomial kernel parameterized by the universe size [23]. We observe thatthe reductions from the H ITTING S ET problem to the O PTIMAL D EFENSE problem in The-orem 15 and ?? are polynomial parameter transformations. Hence we immediately have thefollowing corollary. . Dey, N. Misra, S. Nath, and G. Shakya 17 ◮ Corollary 27.
The O PTIMAL D EFENSE and O PTIMAL A TTACK problems for the scoring rulesand the Condorcet rule do not admit a polynomial kernel parameterized by ( k a , k d ) . We complement the negative results of Observation 1 and Theorem 22 by presenting an FPTalgorithm for the O
PTIMAL D EFENSE problem parameterized by ( k a , k d ) . In the absence ofa defender, that is when k d =
0, Yin et al. [56] showed that the O
PTIMAL D EFENSE problemis polynomial time solvable for the plurality voting rule. Their polynomial time algorithmfor the O
PTIMAL D EFENSE problem can easily be extended to any scoring rule. Using thispolynomial time algorithm, we design the following O ∗ ( k k d a ) time algorithm for the O PTIMAL D EFENSE problem for scoring rules. This result shows that the O
PTIMAL D EFENSE problemis fixed parameter tractable with ( k a , k d ) as the parameter. ◮ Theorem 28.
There is an algorithm for the O PTIMAL D EFENSE problem for every scoringrule and the Condorcet voting rule which runs in time O ∗ ( k k d a ) . Proof.
Let us prove the result for any scoring rule. The proof for the Condorcet voting ruleis exactly similar. Initially we run the attacking algorithm over the n voter groups withoutany group being protected. If a successful attack exists, the algorithm outputs the k a groupsto be deleted. We recursively branch on k a cases by protecting one of these k a groups ineach branch and running the attacking algorithm again. In addition, the parameter k d isalso reduced by 1 each time a group is protected. When k d =0, the attacking algorithm isrun on all the leaves of the tree and a valid protection strategy exists as long as for at leastone of the leaves the attack outputs no i.e. after deploying resources to protect k d groupsthe attacker is unable to change the outcome of the election with any strategy. The groupsto be protected is determined by traversing the tree that leads to the particular leaf whichdid not output an attack. Clearly the number of nodes in this tree is bounded by k k d a . Theamount of time taken to find an attack at each node is bounded by poly ( n ) . Hence therunning time of this algorithm is bounded by k k d a . poly ( n ) . ◭ Though the previous sections show that the optimal defending problem is computationallyintractable, it is a worst-case result. In practice, elections have voting profiles that aregenerated from some (possibly known) distribution. In this section, we conduct an empiricalstudy to understand how simple defending strategies perform for two such statistical votergeneration models. The defending strategies we consider are variants of a simple greedypolicy.
Defending strategy : For a given voting profile and a voting rule, the defending strategy findsthe winner. Suppose the winner is a . The strategy considers a with every other candidate,and for each such pair it creates a sorted list of classes based on the winning margin ofvotes for a in those classes, and picks the top k d classes to form a sub-list. Now, amongall these ( m − ) sorted sub-lists, the strategy picks the most frequent k d classes to protect.We call this version of the strategy GREEDY
1. For certain profiles an optimal attacker (a)may change the outcome by attacking some of the unprotected classes or (b) is unable to G R EE D Y P P P P P P P P PV V V V V V V V VB B B B B B B B B
PluralityVetoBorda k d G R EE D Y optimal and defendedoptimal but not defendednot optimal Figure 1
Performances of
GREEDY
GREEDY change the outcome. If (a) occurs, then there is a possibility that for the value of k d theredoes not exist any defense strategy which can guard the election from all possible strategiesof the attacker. In that case, GREEDY k d protected classes, it is easy to findif there exists an optimal attack strategy, while it is not so easy to identify whether theredoes not exist any defending strategy if the GREEDY
GREEDY
GREEDY k d classes uniformly at random. Call this strategy GREEDY G R EE D Y P P P P P P P P PV V V V V V V V VB B B B B B B B B
PluralityVetoBorda k d G R EE D Y optimal and defendedoptimal but not defendednot optimal Figure 2
Performances of
GREEDY
GREEDY
Voting profile generation : Fix m =
5. We generate 1000 preference profiles over thesealternatives for n = m alternatives. The voters are partitioned into 12classes containing equal number of voters. We consider three voting rules: plurality, veto,and Borda. The lower plot in Figure 1 shows the number of profiles which belongs to thethree categories: (i) GREEDY
GREEDY
GREEDY . Dey, N. Misra, S. Nath, and G. Shakya 19 exists (not optimal). The x-axis shows different values of k d and we fix k a = − k d .The upper plot of Figure 1 shows the fraction of the profiles successfully defended by GREEDY
GREEDY
GREEDY k d classes 100 times. These fractions therefore servesas an empirical probability of successful defense of GREEDY
GREEDY a on top and the strict order of the ( m − ) alternatives is picked uniformly atrandom, a similar 40% profiles with some other alternative b on top, and the remaining20% preferences are picked uniformly at random from the set of all possible strict prefer-ence orders. Similar experiments are run on this generation model and results are shown inFigure 2.The results show that even though optimal defense is a hard problem, a simple strategylike greedy achieves more than 70% optimality. From the rest 30% non-optimal cases, thevariant GREEDY k d = k a =
6. This empirically hints at a possibility that defending real elections may not betoo difficult.
We have considered the O
PTIMAL D EFENSE problem from a primarily parameterized per-spective for scoring rules and the Condorcet voting rule. We showed hardness in the numberof candidates, the number of resources for the defender or the attacker. On the other hand,we show tractability for the combined parameter ( k a , k d ) . We also introduced the O PTIMAL A TTACK problem, which is hard even for the combined parameter ( k a , k d ) , and also showedthe hardness for a constant number of candidates. Even though the O PTIMAL D EFENSE prob-lem is hard, empirically we show that relatively simple mechanisms ensure good defendingperformance for reasonable voting profiles.
References Election day bombings sweep pakistan: Over 30 killed, more than 200 injured. , 2013. Bo An, Matthew Brown, Yevgeniy Vorobeychik, and Milind Tambe. Security games with surveil-lance cost and optimal timing of attack execution. In
International conference on AutonomousAgents and Multi-Agent Systems, AAMAS ’13, Saint Paul, MN, USA, May 6-10, 2013 , pages 223–230, 2013. John J. Bartholdi, Craig A. Tovey, and Michael A. Trick. How hard is it to control an election?
Mathematical and Computer Modelling , 16(8):27 – 40, 1992. Dorothea Baumeister, Magnus Roos, and Jörg Rothe. Computational complexity of two variantsof the possible winner problem. In
The 10th International Conference on Autonomous Agents andMultiagent Systems (AAMAS) , pages 853–860, 2011. Nadja Betzler and Johannes Uhlmann. Parameterized complexity of candidate control in elec-tions and related digraph problems.
Theor. Comput. Sci. , 410(52):5425–5442, 2009. Satarupa Bhattacharjya. Low turnout and invalid votes mark first post war general polls. , 2010. Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel Bounds for Disjoint Cyclesand Disjoint Paths. In Amos Fiat and Peter Sanders, editors,
Proc. 17th Annual European Sym-posium,on Algorithms (ESA 2009), Copenhagen, Denmark, September 7-9, 2009. , volume 5757 of
Lecture Notes in Computer Science , pages 635–646. Springer, 2009. Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel Procaccia. Handbook ofcomputational social choice, 2016. Laurent Bulteau, Jiehua Chen, Piotr Faliszewski, Rolf Niedermeier, and Nimrod Talmon. Com-binatorial voter control in elections.
Theor. Comput. Sci. , 589:99–120, 2015. Jiehua Chen, Piotr Faliszewski, Rolf Niedermeier, and Nimrod Talmon. Elections with few voters:Candidate control can be easy. In
Proc. Twenty-Ninth AAAI Conference on Artificial Intelligence,January 25-30, 2015, Austin, Texas, USA. , pages 2045–2051, 2015. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein.
Introduction toAlgorithms, Third Edition . The MIT Press, 3rd edition, 2009. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilip-czuk, Michal Pilipczuk, and Saket Saurabh.
Parameterized Algorithms . Springer, 2015. Palash Dey. Manipulative elicitation - A new attack on elections with incomplete preferences.
Theor. Comput. Sci. , 731:36–49, 2018. Palash Dey. Optimal bribery in voting. In
Proc. International Conference on Autonomous Agentsand Multiagent Systems, AAMAS , 2019. Palash Dey and Neeldhara Misra. On the exact amount of missing information that makes findingpossible winners hard. In
Proc. 42nd International Symposium on Mathematical Foundations ofComputer Science, MFCS , pages 57:1–57:14, 2017. Palash Dey, Neeldhara Misra, and Y. Narahari. Detecting possible manipulators in elections.In
Proc. International Conference on Autonomous Agents and Multiagent Systems, AAMAS , pages1441–1450, 2015. Palash Dey, Neeldhara Misra, and Y. Narahari. Kernelization complexity of possible winner andcoalitional manipulation problems in voting.
Theor. Comput. Sci. , 616:111–125, 2016. Palash Dey, Neeldhara Misra, and Y. Narahari. Kernelization complexity of possible winner andcoalitional manipulation problems in voting.
Theor. Comput. Sci. , 616:111–125, 2016. Palash Dey, Neeldhara Misra, and Y. Narahari. Frugal bribery in voting.
Theor. Comput. Sci. ,676:15–32, 2017. Palash Dey, Neeldhara Misra, and Y. Narahari. Complexity of manipulation with partial inform-ation in voting.
Theor. Comput. Sci. , 726:78–99, 2018. Palash Dey, Neeldhara Misra, and Y. Narahari. Parameterized dichotomy of choosing committeesbased on approval votes in the presence of outliers.
Theor. Comput. Sci. , 2019. . Dey, N. Misra, S. Nath, and G. Shakya 21 Rod G Downey and Michael Ralph Fellows.
Parameterized Complexity , volume 3. springer Heidel-berg, 1999. Gábor Erdélyi, Edith Hemaspaandra, and Lane A. Hemaspaandra. More natural models of elect-oral control by partition. In
Algorithmic Decision Theory - 4th International Conference, ADT 2015,Lexington, KY, USA, September 27-30, 2015, Proceedings , pages 396–413, 2015. Gábor Erdélyi, Markus Nowak, and Jörg Rothe. Sincere-strategy preference-based approvalvoting fully resists constructive control and broadly resists destructive control.
Math. Log. Q. ,55(4):425–443, 2009. Gábor Erdélyi and Jörg Rothe. Control complexity in fallback voting. In
Theory of Computing2010, CATS 2010, Brisbane, Australia, January 2010 , pages 39–48, 2010. Piotr Faliszewski, Edith Hemaspaandra, and Lane A. Hemaspaandra. Multimode control attackson elections.
J. Artif. Intell. Res. (JAIR) , 40:305–351, 2011. Piotr Faliszewski, Edith Hemaspaandra, and Lane A. Hemaspaandra. Weighted electoral control.
J. Artif. Intell. Res. (JAIR) , 52:507–542, 2015. Piotr Faliszewski, Edith Hemaspaandra, Lane A. Hemaspaandra, and Jörg Rothe. Llull andcopeland voting broadly resist bribery and control. In
Proc. Twenty-Second AAAI Conference onArtificial Intelligence, July 22-26, 2007, Vancouver, British Columbia, Canada , pages 724–730,2007. Piotr Faliszewski, Edith Hemaspaandra, Lane A. Hemaspaandra, and Jörg Rothe. Copelandvoting fully resists constructive control. In
Algorithmic Aspects in Information and Management,4th International Conference, AAIM 2008, Shanghai, China, June 23-25, 2008. Proceedings , pages165–176, 2008. Piotr Faliszewski, Edith Hemaspaandra, Lane A. Hemaspaandra, and Jörg Rothe. Llull and cope-land voting computationally resist bribery and constructive control.
J. Artif. Intell. Res. (JAIR) ,35:275–341, 2009. Piotr Faliszewski, Edith Hemaspaandra, Lane A. Hemaspaandra, and Jörg Rothe. The shieldthat never was: societies with single-peaked preferences are more open to manipulation andcontrol. In
Proc. 12th Conference on Theoretical Aspects of Rationality and Knowledge (TARK-2009), Stanford, CA, USA, July 6-8, 2009 , pages 118–127, 2009. Piotr Faliszewski, Edith Hemaspaandra, Lane A. Hemaspaandra, and Jörg Rothe. The shield thatnever was: Societies with single-peaked preferences are more open to manipulation and control.
Inf. Comput. , 209(2):89–107, 2011. Zack Fitzsimmons, Edith Hemaspaandra, and Lane A. Hemaspaandra. Control in the presence ofmanipulators: Cooperative and competitive cases. In
IJCAI 2013, Proc. 23rd International JointConference on Artificial Intelligence, Beijing, China, August 3-9, 2013 , pages 113–119, 2013. Jörg Flum and Martin Grohe.
Parameterized Complexity Theory , volume 3. Springer, 2006. Edith Hemaspaandra, Lane A. Hemaspaandra, and Jörg Rothe. Hybrid elections broadencomplexity-theoretic resistance to control.
Math. Log. Q. , 55(4):397–424, 2009. Edith Hemaspaandra, Lane A. Hemaspaandra, and Jörg Rothe. Controlling candidate-sequentialelections. In
ECAI 2012 - 20th European Conference on Artificial Intelligence. Including Prestigi-ous Applications of Artificial Intelligence (PAIS-2012) System Demonstrations Track, Montpellier,France, August 27-31 , 2012 , pages 905–906, 2012. Lane A. Hemaspaandra, Rahman Lavaee, and Curtis Menton. Schulze and ranked-pairs votingare fixed-parameter tractable to bribe, manipulate, and control. In
International conference on
Autonomous Agents and Multi-Agent Systems, AAMAS ’13, Saint Paul, MN, USA, May 6-10, 2013 ,pages 1345–1346, 2013. Joshua Letchford, Vincent Conitzer, and Kamesh Munagala. Learning and approximating theoptimal strategy to commit to. In
Algorithmic Game Theory, Second International Symposium,SAGT 2009, Paphos, Cyprus, October 18-20, 2009. Proceedings , pages 250–262, 2009. Hong Liu, Haodi Feng, Daming Zhu, and Junfeng Luan. Parameterized computational complex-ity of control problems in voting systems.
Theor. Comput. Sci. , 410(27-29):2746–2753, 2009. Hong Liu and Daming Zhu. Parameterized complexity of control problems in maximin election.
Inf. Process. Lett. , 110(10):383–388, 2010. Hong Liu and Daming Zhu. Parameterized complexity of control by voter selection in maximin,copeland, borda, bucklin, and approval election systems.
Theor. Comput. Sci. , 498:115–123,2013. Krzysztof Magiera and Piotr Faliszewski. How hard is control in single-crossing elections? In
ECAI 2014 - 21st European Conference on Artificial Intelligence, 18-22 August 2014, Prague, CzechRepublic - Including Prestigious Applications of Intelligent Systems (PAIS 2014) , pages 579–584,2014. Nicholas Mattei, Nina Narodytska, and Toby Walsh. How hard is it to control an election bybreaking ties? In Torsten Schaub, Gerhard Friedrich, and Barry O’Sullivan, editors,
ECAI , volume263 of
Frontiers in Artificial Intelligence and Applications , pages 1067–1068. IOS Press, 2014. Cynthia Maushagen and Jörg Rothe. Complexity of control by partitioning veto and maximinelections and of control by adding candidates to plurality elections. In
ECAI 2016 - 22ndEuropean Conference on Artificial Intelligence, 29 August-2 September 2016, The Hague, The Neth-erlands - Including Prestigious Applications of Artificial Intelligence (PAIS 2016) , pages 277–285,2016. David C McGarvey. A theorem on the construction of voting paradoxes.
Econometrica , pages608–610, 1953. Curtis Menton. Normalized range voting broadly resists control.
Theory Comput. Syst. ,53(4):507–531, 2013. Curtis Glen Menton and Preetjot Singh. Control complexity of schulze voting. In
IJCAI 2013,Proc. 23rd International Joint Conference on Artificial Intelligence, Beijing, China, August 3-9,2013 , pages 286–292, 2013. Tomasz Miasko and Piotr Faliszewski. The complexity of priced control in elections.
Ann. Math.Artif. Intell. , 77(3-4):225–250, 2016. Rolf Niedermeier. Invitation to fixed-parameter algorithms.
Habilitationschrift, University ofTübingen , 2002. David C. Parkes and Lirong Xia. A complexity-of-strategic-behavior comparison betweenschulze’s rule and ranked pairs. In
Proc. Twenty-Sixth AAAI Conference on Artificial Intelligence,July 22-26, 2012, Toronto, Ontario, Canada. , 2012. Tomasz Put and Piotr Faliszewski. The complexity of voter control and shift bribery underparliament choosing rules.
Trans. Computational Collective Intelligence , 23:29–50, 2016. Jianxin Wang, Weimin Su, Min Yang, Jiong Guo, Qilong Feng, Feng Shi, and Jianer Chen.Parameterized complexity of control and bribery for d-approval elections.
Theor. Comput. Sci. ,595:82–91, 2015. . Dey, N. Misra, S. Nath, and G. Shakya 23 Scott Wolchok, Eric Wustrow, Dawn Isabel, and J. Alex Halderman. Attacking the washington,D.C. internet voting system. In
Financial Cryptography and Data Security - 16th InternationalConference, FC 2012, Kralendijk, Bonaire, Februray 27-March 2, 2012, Revised Selected Papers ,pages 114–128, 2012. Lirong Xia and Vincent Conitzer. Determining possible and necessary winners under commonvoting rules given partial orders.
J. Artif. Intell. Res. , 41(2):25–67, 2011. Yue Yin, Yevgeniy Vorobeychik, Bo An, and Noam Hazon. Optimally protecting elections. In